Stochastic Calculus and Applications
A.Y. 2025/2026
Learning objectives
The objective of the course is to provide an introduction to stochastic calculus, more specifically Ito calculus. Starting from the main definitions and results of stochastic processes theory, in particular of the Brownian motion, we will introduce Ito's stochastic integral and we will investigate its main properties. Moreover, we will study the stochastic differential equations, showing existence and uniqueness of their solutions in the Lipschitz case. Finally, we will present some applications in analysis, e.g. Feynman-Kac formula, which highlight the link between stochastic differential equations and partial differential equations.
Expected learning outcomes
The students will learn the notion of stochastic integral, its main properties and some fundamental results of stochastic calculus based on it. Moreover, he/she will be able to solve some classes of stochastic differential equations, to study their main properties also by exploiting the link between them and partial differential equations.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
Brownian motion
- Definition of the Brownian motion
- Wiener measure
- Properties of the Brownian motion: aw of iterated logarithm, reflection principle, infinite variation of trajectories
Stochastic integral and stochastic calculus
- Construction of Itô's stochastic integral and its properties
- Itô's formula
- Multidimensional stochastic integral
- Lévy's characterization of Brownian motion
- Girsanov's theorem
- Brownian martingale representation theorem
Stochastic differential equations
- Definitions of strong/weak solution, definitions of pathwise/in law uniqueness
- A priori estimates
- Existence and uniqueness of strong solutions
- Dependence on initial data
- Markov property
- Weak solutions and Girsanov's theorem
Stochastic differential equations and partial differential equations
- Probabilistic representation for classical solutions of Dirichlet or Cauchy-Dirichlet problems
- Forward and backward Kolmogorov equations
- Other possible applications: convergence to equilibrium, diffusion models in generative AI etc.
- Definition of the Brownian motion
- Wiener measure
- Properties of the Brownian motion: aw of iterated logarithm, reflection principle, infinite variation of trajectories
Stochastic integral and stochastic calculus
- Construction of Itô's stochastic integral and its properties
- Itô's formula
- Multidimensional stochastic integral
- Lévy's characterization of Brownian motion
- Girsanov's theorem
- Brownian martingale representation theorem
Stochastic differential equations
- Definitions of strong/weak solution, definitions of pathwise/in law uniqueness
- A priori estimates
- Existence and uniqueness of strong solutions
- Dependence on initial data
- Markov property
- Weak solutions and Girsanov's theorem
Stochastic differential equations and partial differential equations
- Probabilistic representation for classical solutions of Dirichlet or Cauchy-Dirichlet problems
- Forward and backward Kolmogorov equations
- Other possible applications: convergence to equilibrium, diffusion models in generative AI etc.
Prerequisites for admission
Knowledge of the basis of probability theory (in particular, construction of probability spaces, random vectors, conditional expectation, various types of convergence) and of stochastic processes (in particular martingales). Taking the courses Probability and Advanced Probability is strongly recommended.
Teaching methods
Lectures in classrooms, usually via tablet and projector.
Teaching Resources
- P. Baldi, Stochastic Calculus. An Introduction Through Theory and Exercises, Springer, 2017.
- lecture notes by the teacher
Other references:
- I. Karatzas, S. E. Shevre, Brownian Motion and Stochastic Calculus, second edition, Springer, 1991.
- D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, third edition, Springer, 1999.
- F. Caravenna, Moto browniano e analisi stocastica, 2011.
- lecture notes by the teacher
Other references:
- I. Karatzas, S. E. Shevre, Brownian Motion and Stochastic Calculus, second edition, Springer, 1991.
- D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, third edition, Springer, 1999.
- F. Caravenna, Moto browniano e analisi stocastica, 2011.
Assessment methods and Criteria
The exam consists of an oral examination. Students will be asked to describe and discuss some of the theoretical results of the course and to solve some exercises. The goal is to assess the knowledge and the understanding of the topics treated in the course as well as the ability to put them into context and to correctly apply them.
MAT/06 - PROBABILITY AND STATISTICS - University credits: 9
Practicals: 24 hours
Lessons: 49 hours
Lessons: 49 hours
Professors:
Cosso Andrea, Fuhrman Marco Alessandro
Professor(s)
Reception:
Upon appointment by email
Department of Mathematics, via Saldini 50, office 1027 or on Microsoft Teams
Reception:
Monday, 10:30 am - 1:30 pm (upon appointment, possibly suppressed for academic duties)
Department of Mathematics, via Saldini 50, office 1017.